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Proof step 59 from proof inv-gc-copy (previous step)

# ax = # ndx1, abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax, err = ESUCCESS, fi = fi2[ndx1.keys, ax], ndx0 = nd0 ' + ndx, nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx1], remove-refsize0(gstore1) = gstore1, elems nx1 ∩ jro0 = ∅, (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx ∩ jro = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore3 = ∅, elems nx0 ∩ jro3 = ∅, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3, jro2 < jmaxino3, fi2.inos < jmaxino3, n0 < nd0.flashsize, p_os < pmaxino, fi2[ndx1.keys, ax].inos < jmaxino0, fi1.inos < pmaxino, inv(gstore3[nx.last]), inv(gstore2 \ jro0), inv(gstore1), inv(gstore3 \ jro), inv(gstore \ jro3), key-cons(nd0.key, nd0), key-cons(ndx1), key-cons(ndx), key-cons(ndx3), new(ax, nodes(gstore2 ↓, jro0)), refsize-cons(gstore2, range fi2), refsize-cons(gstore3, range fi2[ndx1.keys, ax]), refsize-cons(gstore1, range fi1), refsize-cons(gstore, range fi2[ndx1.keys, ax]), ri-inj(fi2), ri-inj(fi2[ndx1.keys, ax]), ri-inj(fi1), nx.last ∈ gstore3, nd0.key ∈ fi2, jmaxino3 ≤ jmaxino0, pmaxino ≤ jmaxino3, n0 ≤ LEB_SIZE, nd0.flashsize + ndx.flashsize ≤ LEB_SIZE, pmaxino ≤ jmaxino0, fi2.lebs ⊆ dom gstore, dom gstore2 ⊆ dom gstore, fi2[ndx1.keys, ax].lebs ⊆ dom gstore, fi2.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore2, ndx1.keys.toSet ⊆ dom fi2, ndx.keys.toSet ⊆ dom fi2, ndx3.keys.toSet ⊆ dom fi2, range fi2 ⊆ dom nodes(gstore2 ↓, jro0), range fi2[ndx1.keys, ax] ⊆ dom nodes(gstore3 ↓, jro), dom gstore2 ⊆ dom gstore3, fi2[ndx1.keys, ax].lebs ⊆ dom gstore3, fi1.lebs ⊆ dom gstore1, dom gstore3 ⊆ dom gstore, jvalid1 → valid-journal-head(nx, gstore3, jro), jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), err ≠ ESUCCESS → abs-log(nx0, gstore ↓) = abs-log(nx, gstore3 ↓) ∧ nodes(gstore ↓, jro3) = nodes(gstore3 ↓, jro), err = ESUCCESS → # ax1 = # ndx + 1 ∧ ¬ dups(ax1) ∧ nodes-cons(ax1, nd0 ' + ndx, gstore, jro3) ∧ abs-log(nx0, gstore ↓) = abs-log(nx, gstore3 ↓) + ax1 ∧ nodes(gstore ↓, jro3) = nodes(gstore3 ↓, jro)[ax1, nd0 ' + ndx] ∧ new(ax1, nodes(gstore3 ↓, jro)), jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3), ndx3 = [] ∨ LEB_SIZE < (nd0.flashsize + ndx.flashsize) + ndx3.head.flashsize, err = ESUCCESS ∨ ⌊err⌋ ⊦ ⟪gjournal_gc_update_index#(jvalid0, fi1, jro3, gstore1, nx0, pmaxino, p_os, jvalid; ax1, ndx0, jmaxino0, fi, gstore)⟫ ( (err = ESUCCESS ∨ ⌊err⌋ ) ∧ jmaxino3 ≤ jmaxino0 ∧ (jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3)) ∧ fi.inos < jmaxino0 ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv(gstore \ jro3) ∧ dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3 ∧ ri-inj(fi) ∧ ndx3.keys.toSet ⊆ dom fi ∧ range fi ⊆ dom nodes(gstore ↓, jro3) ∧ dom gstore2 ⊆ dom gstore ∧ (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅ ∧ (∃ ax, ndx0, ndx4. ndx1 + nd0 ' + ndx + ndx3 = ndx0 + ndx4 ∧ (err = ESUCCESS → ndx4 = ndx3) ∧ new(ax, nodes(gstore2 ↓, jro0)) ∧ ¬ dups(ax) ∧ # ax = # ndx0 ∧ fi = fi2[ndx0.keys, ax] ∧ nodes(gstore ↓, jro3) = nodes(gstore2 ↓, jro0)[ax, ndx0] ∧ abs-log(nx0, gstore ↓) = abs-log(nx1, gstore2 ↓) + ax) ∧ ¬ # ndx + # ndx3 < (err = ESUCCESS ⊃ # ndx3;0)), jmaxino3 = 0, nx = [], pmaxino = 0, dups(ndx3.keys), dups(ndx1.keys + ndx.keys), dups(ndx.keys + ndx3.keys), dups(ax), dups(nx), dups(nx1), dups(ndx1.keys + ndx.keys + ndx3.keys), dups(ndx1.keys), dups(ndx.keys), dups(nx0), nd0.key ∈ ndx3.keys, nd0.key ∈ ndx.keys, nd0.key ∈ ndx1.keys

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 60:
# ax1 = # ndx + 1, # ax = # ndx1, abs-log(nx0, gstore ↓) = abs-log(nx, gstore3 ↓) + ax1, abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax, fi = fi2[ndx1.keys, ax], ndx0 = nd0 ' + ndx, nodes(gstore ↓, jro3) = nodes(gstore3 ↓, jro)[ax1, nd0 ' + ndx], nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx1], remove-refsize0(gstore1) = gstore1, elems nx1 ∩ jro0 = ∅, (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx ∩ jro = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore3 = ∅, elems nx0 ∩ jro3 = ∅, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3, jmaxino3 ≠ 0, nx ≠ [], pmaxino ≠ 0, jro2 < jmaxino3, fi2.inos < jmaxino3, n0 < nd0.flashsize, p_os < pmaxino, fi2[ndx1.keys, ax].inos < jmaxino0, fi1.inos < pmaxino, inv(gstore3[nx.last]), inv(gstore2 \ jro0), inv(gstore1), inv(gstore3 \ jro), inv(gstore \ jro3), key-cons(nd0.key, nd0), key-cons(ndx1), key-cons(ndx), key-cons(ndx3), new(ax1, nodes(gstore3 ↓, jro)), new(ax, nodes(gstore2 ↓, jro0)), nodes-cons(ax1, nd0 ' + ndx, gstore, jro3), refsize-cons(gstore2, range fi2), refsize-cons(gstore3, range fi2[ndx1.keys, ax]), refsize-cons(gstore1, range fi1), refsize-cons(gstore, range fi2[ndx1.keys, ax]), ri-inj(fi2), ri-inj(fi2[ndx1.keys, ax]), ri-inj(fi1), nx.last ∈ gstore3, nd0.key ∈ fi2, jmaxino3 ≤ jmaxino0, pmaxino ≤ jmaxino3, n0 ≤ LEB_SIZE, nd0.flashsize + ndx.flashsize ≤ LEB_SIZE, pmaxino ≤ jmaxino0, fi2.lebs ⊆ dom gstore, dom gstore2 ⊆ dom gstore, fi2[ndx1.keys, ax].lebs ⊆ dom gstore, fi2.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore2, ndx1.keys.toSet ⊆ dom fi2, ndx.keys.toSet ⊆ dom fi2, ndx3.keys.toSet ⊆ dom fi2, range fi2 ⊆ dom nodes(gstore2 ↓, jro0), range fi2[ndx1.keys, ax] ⊆ dom nodes(gstore3 ↓, jro), dom gstore2 ⊆ dom gstore3, fi2[ndx1.keys, ax].lebs ⊆ dom gstore3, fi1.lebs ⊆ dom gstore1, dom gstore3 ⊆ dom gstore, ¬ dups(ax1), ¬ dups(ndx3.keys), ¬ dups(ndx1.keys + ndx.keys), ¬ dups(ndx.keys + ndx3.keys), ¬ dups(ax), ¬ dups(nx), ¬ dups(nx1), ¬ dups(ndx1.keys + ndx.keys + ndx3.keys), ¬ dups(ndx1.keys), ¬ dups(ndx.keys), ¬ dups(nx0), ¬ nd0.key ∈ ndx3.keys, ¬ nd0.key ∈ ndx.keys, ¬ nd0.key ∈ ndx1.keys, jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3), jvalid1 → valid-journal-head(nx, gstore3, jro), jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), ndx3 = [] ∨ LEB_SIZE < (nd0.flashsize + ndx.flashsize) + ndx3.head.flashsize ⊦ ⟪gjournal_gc_update_index#(jvalid0, fi1, jro3, gstore1, nx0, pmaxino, p_os, jvalid; ax1, ndx0, jmaxino0, fi, gstore)⟫ ( jmaxino3 ≤ jmaxino0 ∧ (jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3)) ∧ fi.inos < jmaxino0 ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv(gstore \ jro3) ∧ dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3 ∧ ri-inj(fi) ∧ ndx3.keys.toSet ⊆ dom fi ∧ range fi ⊆ dom nodes(gstore ↓, jro3) ∧ dom gstore2 ⊆ dom gstore ∧ (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅ ∧ (∃ ax0. new(ax0, nodes(gstore2 ↓, jro0)) ∧ ¬ dups(ax0) ∧ # ax0 = # ndx1 + # ndx + 1 ∧ fi = fi2[ndx1.keys + nd0.key ' + ndx.keys, ax0] ∧ nodes(gstore ↓, jro3) = nodes(gstore2 ↓, jro0)[ax0, ndx1 + nd0 ' + ndx] ∧ abs-log(nx0, gstore ↓) = abs-log(nx1, gstore2 ↓) + ax0))

Rule argument

Additional information

Used simplifier rules: ⊦ nd ∈ nd0 ' + ndx ↔ nd = nd0 ∨ nd ∈ ndx; (s)%in-14: ⊦ nd ' + ndx = ndx0 → nd ∈ ndx0; (forward) from specification list[node-list]%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ ¬ m + n < m; (s)% ⊦ key + kx = key ' + kx; (s)% ⊦ (nd ' + ndx).keys = nd.key + ndx.keys; (s)% ⊦ (ndx0 + ndx1).keys = ndx0.keys + ndx1.keys; (s)% ⊦ m +1 = m + 1; (s)% ⊦ #(nd ' + ndx) = (# ndx)+1; (s)% ⊦ #(ndx + ndx0) = # ndx + # ndx0; (s)%a: ⊦ (ndx + ndx0) + ndx1 = ndx + ndx0 + ndx1; from specification list[node-list]% ⊦ ndx + ndx0 = ndx1 + ndx0 ↔ ndx = ndx1; (s)